Black & Litterman Estimator

Data

from plotly.io import show
from sklearn.model_selection import train_test_split

from skfolio import Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import MeanRisk, ObjectiveFunction
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import BlackLitterman

prices = load_sp500_dataset()
X = prices_to_returns(prices)
X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False)

X_train.head()

	AAPL	AMD	BAC	BBY	CVX	GE	HD	JNJ	JPM	KO	LLY	MRK	MSFT	PEP	PFE	PG	RRC	UNH	WMT	XOM
Date
1990-01-03	0.007576	-0.030303	0.008045	0.118056	-0.016229	-0.001876	0.003581	0.004072	0.033589	-0.014318	0.000000	0.015896	0.005208	-0.009709	0.002938	-0.001813	0.0	-0.019355	0.000000	-0.010079
1990-01-04	0.003759	-0.015500	-0.021355	-0.012422	-0.012831	-0.005639	0.006244	0.002028	0.003991	-0.004993	-0.005557	-0.015647	0.028497	-0.009804	0.016602	-0.019725	0.0	-0.009868	-0.005201	-0.009933
1990-01-05	0.003745	-0.031996	-0.021821	0.000000	-0.014855	-0.009452	-0.013298	-0.010408	0.003975	-0.008212	-0.010874	-0.020641	-0.025189	-0.013991	-0.008646	-0.018004	0.0	-0.043189	-0.010732	-0.005267
1990-01-08	0.003731	0.000000	0.005633	-0.075472	0.009424	0.005725	-0.009883	0.016944	0.000000	0.021159	0.000000	0.012839	0.015504	0.018118	-0.008721	0.018334	0.0	-0.020833	0.013630	0.015381
1990-01-09	-0.007435	0.016527	0.000000	0.000000	-0.007469	-0.020803	-0.026316	-0.031026	-0.031957	-0.007658	-0.011147	-0.007893	-0.002545	-0.013722	-0.021505	0.000000	0.0	-0.024823	-0.026619	-0.020114

Prior View

prior_view = [
    "AAPL == 0.003"
]

Black-Litterman Model

Maximum Sharpe Ratio with Black-Litterman Estimator

max_sharpe_model_bl = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    prior_estimator=BlackLitterman(views=prior_view),  # Impose prior
    portfolio_params=dict(name="Black-Litterman")
)
max_sharpe_model_bl.fit(X_train)
max_sharpe_model_bl.weights_

array([0.62267776, 0.01982462, 0.01976329, 0.01981544, 0.01988257,
       0.01986179, 0.01986453, 0.01987279, 0.01978561, 0.0198974 ,
       0.01988095, 0.01988681, 0.01988106, 0.01989788, 0.01988813,
       0.01988705, 0.0198046 , 0.01985367, 0.01988082, 0.01989322])

Create a vanilla Maximum Sharpe Ratio model.

benchmark_model = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    portfolio_params=dict(name="Empirical")
)
benchmark_model.fit(X_train)
benchmark_model.weights_

array([9.43631399e-02, 1.13184579e-06, 5.04970598e-07, 1.20834667e-01,
       3.18126275e-02, 8.57806907e-07, 7.11596802e-04, 1.24104939e-01,
       9.49223801e-07, 2.77547553e-02, 1.23409042e-06, 1.37593860e-06,
       1.16299875e-01, 5.73516411e-02, 9.58498590e-06, 1.09493919e-01,
       8.64761638e-02, 1.83992252e-01, 1.32350165e-02, 3.35537683e-02])

Prediction

pred_black_litterman = max_sharpe_model_bl.predict(X_train)
pred_benchmark = benchmark_model.predict(X_train)

Analysis

population = Population([pred_black_litterman, pred_benchmark])
population.plot_cumulative_returns()

population.plot_composition()

population.plot_returns_distribution()